On the unimodality of nearly well-dominated trees

Abstract

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of dominating sets of each cardinality in $G$. In \cite{IntroDomPoly2014} Alikhani and Peng conjectured that all domination polynomials are unimodal. In this paper we show that not all trees have log-concave domination polynomial. We also give non-increasing and non-decreasing segments of coefficents in trees. This allows us to show the domination polynomial trees with $Γ(T)-γ(T)<3$ are unimodal.

Figures (3)

• Figure 2.1: The tree $T_k$
• Figure 4.1: Example of Algorithm \ref{['alg:MakeMinimal']}
• Figure 4.2: Example of Algorithm \ref{['alg:a2_to_n2_X']}

Theorems & Definitions (27)

• Proposition 1
• proof
• Lemma 3
• Lemma 5
• proof
• Theorem 6
• proof
• Lemma 7
• proof
• Lemma 9